Remark 3.9 We will not need it in what follows, so only briefly comment here that it is possible to specify precisely the type \((\mathcal {T}^{\prime },n)\)  of the germ of \(L_\mathcal {T}\)  at each point \(\lambda \in L_\mathcal {T}\) . Following [1]} the underlying tree \(T^{\prime }\)  is a canonically defined subquotient of \(T\) , in other words, a diagram \(T^{\prime } \leftarrow S \rightarrow T\) , where \(S\rightarrow T\)  is a full subtree, and \(S\rightarrow T^{\prime }\)  contracts some edges; conversely, any such subquotient can occur. Furthermore, if we partially order \(T\)  with the root \(\rho \in T\)  as minimum, then the root \(\rho ^{\prime }\in T^{\prime }\)  is the unique minimum of the natural induced partial order on \(T^{\prime }\) . Finally, to equip \(T^{\prime }\)  with signs, we restrict the signs of \(T\)  to the subtree \(S\) , then push them forward to \(T^{\prime }\)  using that each edge of \(T^{\prime }\)  is the image of a unique edge of \(S\) .

